To measure the tree-likeness of a directed acyclic graph (DAG), a new width parameter that considers the directions of the arcs was recently introduced: scanwidth . We present the first algorithm that efficiently computes the exact scanwidth of general DAGs. For DAGs with one root and scanwidth k it runs in O ( k ⋅ n k ⋅ m ) time. The algorithm also functions as an FPT algorithm with complexity O ( 2 4 ℓ − 1 ⋅ ℓ ⋅ n + n 2 ) for phylogenetic networks of level- ℓ , a type of DAG used to depict evolutionary relationships among species. Our algorithm performs well in practice, being able to compute the scanwidth of synthetic networks up to 30 reticulations and 100 leaves within 500 seconds. Furthermore, we propose a heuristic that obtains an average practical approximation ratio of 1.5 on these networks. While we prove that the scanwidth is bounded from below by the treewidth of the underlying undirected graph, experiments suggest that for networks the parameters are close in practice.

Semi-directed networks are partially directed graphs that model evolution where the directed edges represent reticulate evolutionary events. We present an algorithm that reconstructs binary n -leaf semi-directed level-1 networks in O ( n 2 ) time from its quarnets (4-leaf subnetworks). Our method assumes we have direct access to all quarnets, yet uses only an asymptotically optimal number of O ( n log ⁡ n ) quarnets. When the network is assumed to contain no triangles, our method instead relies only on four-cycle quarnets and the splits of the other quarnets. A variant of our algorithm works with quartets rather than quarnets and we show that it reconstructs most of a semi-directed level-1 network from an asymptotically optimal O ( n log ⁡ n ) of the quartets it displays. Additionally, we provide an O ( n 3 ) time algorithm that reconstructs the tree-of-blobs of any binary n -leaf semi-directed network with unbounded level from O ( n 3 ) splits of its quarnets.